Q. 3.109 - Elementary Statistics

Question

A quantitative data set has mean $25$ and standard deviation $5$.Fill in the following blanks:

(a)At least $89\%$ of the observation lie between ____ and____ .

(b)At least____ $\%$ of the observation lie between $15$ and $35$.  

Step-by-Step Solution

Answer

(a) $89\%$of the observations lie between $10$ and $40$.

(b)$75\%$ of the observation lie between $15$ and $35$. 

Part(a) Step1: Given information.

We have been given that,

Mean= $25$ , Standard deviation = $5$

We need to find the values where in between $89\%$ of the observations lies.

Part(a) Step2: Simplify.

We know that,

Chebyshev's Rule: At least $100\left(1-\frac{1}{K^2}\right)\%$ of the data values is written $K$ standard deviation from the mean $\left(K>1\right)$.

By using Chebyshev's Rule with $k=3$, we know that at least

                                              $$\begin{gathered} =100\left(1-\frac{1}{K^2}\right)\% \\ =100\left(1-\frac{1}{3^2}\right)\% \\ =100\left(1-\frac{1}{9}\right)\% \\ =100\times \frac{8}{9} \%=89\% \end{gathered}$$

is within $3$ standard deviation from the mean.

Determine the values that are $3$ standard deviation from the mean:

            $$\begin{gathered} Mean-3 S\tan dard deviation =25-3\times 5=25-15=10 \\ Mean+3 S\tan dard deviation =25+3\times 5 = 25+15=40 \end{gathered}$$                                   

Therefore, at least $89\%$ of the observations lie between $10$ and $40$.

Part(b) Step1: Given information.

We have been given that,

Range of the observations is $15$ and $35$ .

We need to find the $\%$ of the observations that lie between given two values.

Part(b) Step2: Simplify.

From the question,$15$ and $35$ are $2$ standard deviation from the mean.

Using Chebyshev's Rule with $k=2$ , we know that at least

                                        $$\begin{gathered} =100\left(1-\frac{1}{K^2}\right)\% \\ =100\left(1-\frac{1}{2^2}\right)\% \\ =100\left(1-\frac{1}{4}\right)\% \\ =100\times \frac{3}{4}\% \\ =75\% \end{gathered}$$

is within $2$ standard deviation from the mean.